The effective resistance of a resistive mesh is a commonly used analogy for various scientific and engineering problems such as voltage drop estimation, substrate coupled noise, distributed control including time synchronization and sensor localization, determining the chemical distance among multiple bonds, and finding the distance between two vertices in a graph. Furthermore, resistive mesh networks are a commonly used structure in electronics to model different elements of an integrated circuit, such as a physical substrate, an integrated circuit layout, and a power distribution network. In particular, for an integrated circuit (“IC”), the electromagnetic interactions above the substrate can be described in a full-wave partial element equivalent circuit (“PEEC”) fashion. However, detailed analysis of the substrate structures, including substrate contacts, guard rings, diffusion layers, and other non-uniform materials, require three-dimensional (“3-D”) simulations of the substrate to achieve the required accuracy.
Due to their versatility, the finite difference method (“FDM”), the finite element method (“FEM”), and other methods of moments are popular methods for obtaining a mesh model (also referred to as a mesh) to represent electromagnetic circuits, including a substrate. However, for substrate modeling, these methods are not without their own serious challenges. For instance in FDM, FDM uses a finite difference basis to generate a uniform discretized mesh, which incurs substrate over-meshing. In addition, FDM yields unnecessarily dense grids far from contact sources. Thus, although FDM leads to a straightforward resistor network, the resulting matrix sizes are prohibitive due to the substrate size, which becomes a bottleneck in implementing the finite difference method. The finite element method does not require uniform meshes. Instead, FEM uses a finite element basis that requires a full 3-D basis, such as a tetrahedral, which leads to additional complexity.
These difficulties have led to the boundary element method (“BEM”). In BEM, only the contact surfaces need to be meshed, leading to far fewer unknowns than the other methods. BEM also requires that Green's function, which satisfies Poisson's equation for the substrate coupling problem, be determined. By enforcing the continuity of potential and normal current flow across layer boundaries, Green's function can be obtained analytically for cases with uniformity.
Generally, the evaluation of Green's function was limited to the top layer of a substrate. Thus, it allowed the contacts to be located only on the top substrate surface. In later developments, the evaluation of the Green's function was extended to handle arbitrarily located contacts in the substrate with the same boundary condition by enabling Green's function evaluation in any layer. In addition, Green's function can be applied to a two-layer substrate with perfect magnetic wall boundary conditions and to a multilayered substrate lying on the ground plane with perfect magnetic sidewalls and top surface.
However, even with the advancements in applying Green's function, the numerical computations of Green's function impose difficult challenges. If the substrate is given with different boundary conditions or has irregular layers with arbitrary variations in conductivity, a complete solution to derive the analytic Green's function does not currently exist. To overcome this, attempts were made to reduce the cost of FEM and FDM by eliminating a substantial fraction of the internal nodes, also referred to as network reduction. Examples of such network reduction techniques to simplify RC networks are a congruence transform, Voronoi polygons, and a combined BEM/FEM based network reduction method.
As a mixed technique, the combined BEM/FEM method solves separate problems in their own domains. In addition, BEM and FEM results are combined via the potentials in the interface nodes to take advantage of the versatility of BEM and FEM (which is cheaper for a regular structure). For the meshing requirement, the BEM mesh should be the dual of the triangular FEM mesh to enforce the constraint that the node along the interface is aligned. In practice, it is very difficult to enforce this meshing requirement since the nodes along the interface are typically not aligned. Current art does not provide any means to align the nodes for a more accurate mesh.
Therefore, it is desirable to present novel methods for electromagnetic circuit modeling, which corrects for the mismatched interface nodes in a mesh and reduces the amount of complexity of the mesh.